System and method for progressive stereo matching of digital images

ABSTRACT

A method and a system for progressive stereo matching of digital images representing a scene. In general, the present invention uses a progressive iterative technique that includes a disparity gradient limit principle and a least commitment strategy. Generally unambiguous pixel matches are found by beginning with a few reliable pixel matches and finding progressively more unambiguous pixel matches. Unambiguous pixel matches are used to define the search ranges for each pixel to guide matching in the current iteration. Unambiguous pixel matches then are found using a novel correlation technique and based on a correlation score associated with a pixel match. The search range is capable of being rotated, and is part of a novel correlation technique that provides a more robust estimate of pixel match reliability. Potential pixel matches found in the search ranges are tested for ambiguity and unambiguous matches are selected and added to the reliable pixel matches.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates in general to computer vision and moreparticularly to a method and a system for progressive stereo matching ofdigital images that represent a scene.

2. Related Art

Stereo matching of digital images is widely used in many computer visionapplications (such as, for example, fast object modeling and prototypingfor computer-aided drafting (CAD), object segmentation and detection forhuman-computer interaction (HCI), video compression, and visualsurveillance) to provide three-dimensional (3-D) depth information.Stereo matching obtains images of a scene from two or more cameraspositioned at different locations and orientations in the scene. Thesedigital images are obtained from each camera at approximately the sametime and points in each of the image are matched corresponding to a 3-Dpoint in space. In general, points from different images are matched bysearching a portion of the images and using constraints (such as anepipolar constraint) to correlate a point in one image to a point inanother image.

Several types of matching techniques exist and generally may beclassified into two broad categories: feature matching and templatematching. Feature matching extracts salient features from the images(such as corners and edge segments) and matches these features acrosstwo or more views. One disadvantage of feature matching, however, isthat only a small subset of image pixels is used to match features. Thismeans that if image pixels used in the matching process are unreliablethen only a coarse and inaccurate 3-D representation of the actual sceneis produced. Template matching uses an assumption that portions ofimages have some similarity and attempts to correlate these similaritiesacross views. Although this assumption may be valid for relativelytextured portions of an image and for image pairs having only smalldifferences, the assumption may lead to unreliable matching at occlusionboundaries and within featureless regions of an image. In addition,template matching yields many false matches due in part to theunreliability of matched image pixels.

These and other existing stereo matching techniques have thedisadvantage of having difficulty specifying an appropriate search rangeand being unable to adapt the search range depending on the observedscene structure. Existing dense matching techniques use the same searchrange (such as a small search window) for the entire image and thus mayyield many false matches. In addition, some matching techniques consideronly one match at a time and propagate the match in a small area, whichalso has the disadvantage of yield many false matches.

Accordingly, what is needed is a method and system for stereo matchingthat is capable of reducing false matches by adapting a search rangedepending on the scene structure. Further, this method and system wouldstart with reliable pixel matches and progressively add reliable andunambiguous matches. What is also needed is a method and system forstereo matching that uses an adaptable search range that is dynamicallydetermined by unambiguous pixel matches. Whatever the merits of theabove-mentioned systems and methods, they do not achieve the benefits ofthe present invention.

SUMMARY OF THE INVENTION

To overcome the limitations in the prior art as described above andother limitations that will become apparent upon reading andunderstanding the present specification, the present invention includesa method and system for performing progressive stereo matching ofdigital images. In particular, the present invention uses a progressiveiterative technique that uses a disparity gradient limit principle and aleast commitment strategy to determine generally unambiguous pixelmatches. The progressive iterative method of the present inventionbegins with a few reliable pixel matches and during successiveiterations finds progressively more unambiguous pixel matches based on anovel correlation technique and a correlation score associated with apixel match. Unambiguous pixel matches found in a previous iteration areused to define search ranges for subsequent pixels to reliably andefficiently determine matches in a current iteration. The presentinvention eliminates the problems of dense matching techniques wherebythe same search range is used for the entire image. In the presentinvention, potentially false matches are reduced by dynamically changingthe search range based on unambiguous matches found in previousiterations.

The present invention provides reliable and accurate pixel matching thatbegins with a reliable set of pixel matches and quickly adds a greaternumber of reliable pixel matches based on a pixel correlation. Inaddition, because at each step of the process the pixel matches areunambiguous, the present invention is highly efficient and fast becausebacktracking due to unreliable pixel matches is eliminated. Unlike manyprior stereo matching techniques, the present invention works well withimages containing featureless regions and is capable of changing anorientation of a search range based on the current set of reliable pixelmatches.

In general, the method of the present invention includes determining aset of reliable (or unambiguous) pixel matches and progressively addinga greater number of unambiguous pixel matches to the set. The initialset of reliable pixel matches may be obtained either automatically(using a suitable matching technique that yields reliable pixel matches)or manually (such as through user input). Pixel matches are sought bydefining a search range for each image and examining pixels within thesearch ranges for potential matches. The search range is capable ofbeing rotated, and is part of a novel correlation technique of thepresent invention that provides a more robust estimate of pixel matchreliability. After an iteration the set of reliable pixel matches actsas a constraint and helps define a new search range. This dynamic searchrange reduces potential matching ambiguities and increases therobustness of the process. Potential pixel matches found in the searchranges are tested for ambiguity and any unambiguous matches are selectedand added to the set of reliable pixels matches. The ambiguity testingincludes determining a correlation score for the pixel match andclassifying the match based on the correlation score. The presentinvention also includes a system for progressive image matching thatincorporates the method of the present invention.

Other aspects and advantages of the present invention as well as a morecomplete understanding thereof will become apparent from the followingdetailed description, taken in conjunction with the accompanyingdrawings, illustrating by way of example the principles of theinvention. Moreover, it is intended that the scope of the invention belimited by the claims and not by the preceding summary or the followingdetailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention can be further understood by reference to thefollowing description and attached drawings that illustrate thepreferred embodiments. Other features and advantages will be apparentfrom the following detailed description of the invention, taken inconjunction with the accompanying drawings, which illustrate, by way ofexample, the principles of the present invention.

Referring now to the drawings in which like reference numbers representcorresponding parts throughout:

FIG. 1 is a block diagram illustrating an apparatus for carrying out thepresent invention.

FIG. 2 is a general block diagram illustrating a system incorporatingthe stereo matching module of the present invention for computer visionapplications.

FIG. 3 is a general block diagram of the stereo matching module of thepresent invention shown in FIG. 2.

FIG. 4 is a general flow diagram of the operation of the presentinvention.

FIG. 5 is a general flow diagram of the operation of the search rangemodule shown in FIG. 3.

FIG. 6 is a detailed flow diagram illustrating a preferred embodiment ofthe present invention.

FIG. 7 illustrates the novel correlation technique between two images ofthe present invention.

FIG. 8 is a working example of FIG. 4 whereby stereo matching of imagesrepresenting a scene is performed in accordance with the presentinvention.

FIGS. 9A-9D illustrate the standard deviations of the predicteddisparities for various iterations of the working example of FIG. 8.

FIG. 10 illustrates a view of a 3-D Euclidean reconstruction of thescene of the working example of FIG. 8.

DETAILED DESCRIPTION OF THE INVENTION

In the following description of the invention, reference is made to theaccompanying drawings, which form a part thereof, and in which is shownby way of illustration a specific example whereby the invention may bepracticed. It is to be understood that other embodiments may be utilizedand structural changes may be made without departing from the scope ofthe present invention.

I. Exemplary Operating Environment

FIG. 1 and the following discussion are intended to provide a brief,general description of a suitable computing environment in which theinvention may be implemented. Although not required, the invention willbe described in the general context of computer-executable instructions,such as program modules, being executed by a computer. Generally,program modules include routines, programs, objects, components, datastructures, etc. that perform particular tasks or implement particularabstract data types. Moreover, those skilled in the art will appreciatethat the invention may be practiced with a variety of computer systemconfigurations, including personal computers, server computers,hand-held devices, multiprocessor systems, microprocessor-based orprogrammable consumer electronics, network PCs, minicomputers, mainframecomputers, and the like. The invention may also be practiced indistributed computing environments where tasks are performed by remoteprocessing devices that are linked through a communications network. Ina distributed computing environment, program modules may be located onboth local and remote computer storage media including memory storagedevices.

With reference to FIG. 1, an exemplary system for implementing theinvention includes a general-purpose computing device in the form of aconventional personal computer 100, including a processing unit 102, asystem memory 104, and a system bus 106 that couples various systemcomponents including the system memory 104 to the processing unit 102.The system bus 106 may be any of several types of bus structuresincluding a memory bus or memory controller, a peripheral bus, and alocal bus using any of a variety of bus architectures. The system memoryincludes read only memory (ROM) 110 and random access memory (RAM) 112.A basic input/output system (BIOS) 114, containing the basic routinesthat help to transfer information between elements within the personalcomputer 100, such as during start-up, is stored in ROM 110. Thepersonal computer 100 further includes a hard disk drive 116 for readingfrom and writing to a hard disk, not shown, a magnetic disk drive 118for reading from or writing to a removable magnetic disk 120, and anoptical disk drive 122 for reading from or writing to a removableoptical disk 124 such as a CD-ROM or other optical media. The hard diskdrive 116, magnetic disk drive 128 and optical disk drive 122 areconnected to the system bus 106 by a hard disk drive interface 126, amagnetic disk drive interface 128 and an optical disk drive interface130, respectively. The drives and their associated computer-readablemedia provide nonvolatile storage of computer readable instructions,data structures, program modules and other data for the personalcomputer 100.

Although the exemplary environment described herein employs a hard disk,a removable magnetic disk 120 and a removable optical disk 124, itshould be appreciated by those skilled in the art that other types ofcomputer readable media that can store data that is accessible by acomputer, such as magnetic cassettes, flash memory cards, digital videodisks, Bernoulli cartridges, random access memories (RAMs), read-onlymemories (ROMs), and the like, may also be used in the exemplaryoperating environment.

A number of program modules may be stored on the hard disk, magneticdisk 120, optical disk 124, ROM 110 or RAM 112, including an operatingsystem 132, one or more application programs 134, other program modules136 and program data 138. A user (not shown) may enter commands andinformation into the personal computer 100 through input devices such asa keyboard 140 and a pointing device 142. In addition, a camera 143 (orother types of imaging devices) may be connected to the personalcomputer 100 as well as other input devices (not shown) including, forexample, a microphone, joystick, game pad, satellite dish, scanner, orthe like. These other input devices are often connected to theprocessing unit 102 through a serial port interface 144 that is coupledto the system bus 106, but may be connected by other interfaces, such asa parallel port, a game port or a universal serial bus (USB). A monitor146 or other type of display device is also connected to the system bus106 via an interface, such as a video adapter 148. In addition to themonitor 146, personal computers typically include other peripheraloutput devices (not shown), such as speakers and printers.

The personal computer 100 may operate in a networked environment usinglogical connections to one or more remote computers, such as a remotecomputer 150. The remote computer 150 may be another personal computer,a server, a router, a network PC, a peer device or other common networknode, and typically includes many or all of the elements described aboverelative to the personal computer 100, although only a memory storagedevice 152 has been illustrated in FIG. 1. The logical connectionsdepicted in FIG. 1 include a local area network (LAN) 154 and a widearea network (WAN) 156. Such networking environments are commonplace inoffices, enterprise-wide computer networks, intranets and the Internet.

When used in a LAN networking environment, the personal computer 100 isconnected to the local network 154 through a network interface oradapter 158. When used in a WAN networking environment, the personalcomputer 100 typically includes a modem 160 or other means forestablishing communications over the wide area network 156, such as theInternet. The modem 160, which may be internal or external, is connectedto the system bus 106 via the serial port interface 144. In a networkedenvironment, program modules depicted relative to the personal computer100, or portions thereof, may be stored in the remote memory storagedevice 152. It will be appreciated that the network connections shownare exemplary and other means of establishing a communications linkbetween the computers may be used.

II. Introduction

The method and system of the present invention includes a progressivematching technique that uses a disparity gradient limit principle and aleast commitment strategy to provide reliable and efficient imagematching. The disparity gradient limit principle states that a disparityshould vary smoothly throughout an image, and that disparity gradientshould not exceed a certain threshold. The least commitment strategystates that the most reliable pixel matches should be selected first anddecisions about unreliable pixel matches should be deferred until themore confidence can be achieved that the pixel match is reliable. Thetwo approaches, along with a correlation approach that, in a preferredembodiment, uses a Bayesian inference framework, are used to providereliable, efficient and rapid matching of images.

In general, the present invention includes an iterative matchingtechnique that may be embodied in a computer hardware or software systemthat progressively matches digital images representing a scene to obtain3-D structure information about the scene. The iterative matchingprocess begins with a few reliable pixel matches that have been obtainedin a suitable manner (such as from user input). For each iterationadditional reliable pixel matches are added to the initial set of pixelmatches according to the least commitment strategy so that the number ofunmatched pixel is rapidly reduced. The set of reliable pixel matchesconstrains a search range for neighboring pixels according to thedisparity gradient limit principle, thereby reducing potential matchingambiguities of the neighbors. A novel correlation technique allows acorrelation window representing the search range to be rotated toprovide a more robust estimate of the reliability of a pixel match.

III. General Overview

As shown in FIGS. 2-10 for the purposes of illustration, the inventionis embodied in a method and a system for progressive stereo matching ofdigital images that represent a scene. FIG. 2 is a general block diagramillustrating a system incorporating the stereo matching module of thepresent invention for computer vision applications. The systemillustrated is only one example of several systems that couldincorporate the progressive stereo matching method and system of thepresent invention. In general, the system 200 receives an input ofdigital images depicting a scene, shown in FIG. 2 as image (1) 210,image (2) 220 through image (N) 230. Each of the images 210, 220, 230may be graphical images captured by a camera that containtwo-dimensional (2-D) image data of the scene, in any suitable format,such as a bitmap or raster image data format.

The system 200 includes a stereo matching module 240 that receives theimages 210, 220, 230 and correlates 2-D information within each of theimages 210, 220, 230 to a three-dimensional point in space. The stereomatching module outputs information (such as depth information) aboutthe 3-D structure of the scene (box 250). This 3-D structure informationmay be used by computer vision applications 260 in any suitable mannersuch as, for example, to digitally reconstruct the scene for stereovision applications.

The present invention reliably and accurately determines a 3-D structureof a scene using at least two input images. The present invention solvesthe problems discussed above of prior matching techniques by startingwith a few reliable initial point matches and progressively adding newpixel matches during an iterative matching process. At each iterationonly reliable and unambiguous matches are selected and added to a set ofreliable pixel matches. In addition, the robustness of the presentinvention is increased by using a correlation match measure that allowsrotation of the match template used to search for pixel matches.

IV. Component Overview

FIG. 3 is a general block diagram of the stereo matching module 240 ofthe present invention shown in FIG. 2. The stereo matching module 240illustrates a single iteration of the module 240. In general, the stereomatching module 240 includes an initialization module 310, which obtainsreliable initial pixel matches and a search range module 320, whichdetermines a search range to look for candidate pixel matches. Moreover,the stereo matching module 240 includes a correlation module 330, forcomputing a correlation score between pixel matches and a classificationmodule 340, for classifying a pixel match based on its correlationscore.

The stereo matching module 240 receives as input multiple images (box350) representing a scene. At least two images are needed as input, andthe multiple images may be obtained in any suitable manner, such as froma camera. The initialization module 310 obtains a few reliable seedpixel matches. These initial matches may be obtained eitherautomatically or through user input. The initial pixel matches are sentto the search range module 320 that defines a search range based on thepixel matches. Pixels from each image within the search range arecorrelated by the correlation module 330. Based on how well the pixelscorrelate, the classification module 340 assign a match label to thepixel match to indicate a pixel match, no pixel match or an unknownmatch at this time. Once enough pixel matches have been found (or amaximum number of iterations has occurred), the stereo matching module240 outputs 3-D image information (box 360) regarding the 3-D structureof the scene.

V. Details of the Components and Operation

The present invention uses a progressive technique to match pixels fromdifferent images and makes use of a disparity gradient strategy and aleast commitment strategy. Based on these two approaches, the presentinvention provides a iterative progressive technique that starts with afew reliable seed matches and then finds progressively more pixelmatches based on these two approaches. The following discussiondescribes these two approaches.

1. Disparity Gradient Limit Principle

Disparity is a parameter that is directly related to depth. Accordingly,a change in disparity coincides with a change in depth. The disparitygradient limit principle states that the disparity should vary smoothlyalmost everywhere, and the disparity gradient should not exceed acertain value. Psychophysical studies have shown that in order for thehuman visual system to binocularly fuse two dots of a simple stereogram,a disparity gradient (the ratio of the disparity difference) between thedots to their cyclopean separation must not exceed a limit of 1. Objectsin the real-world are usually bounded by continuous opaque surfaces, andthe disparity gradient can be considered as a measure of continuity. Thedisparity gradient limit principle provides a constraint on scenejaggedness embracing simultaneously the ideas of opacity, scenecontinuity, and continuity between views. As explained further below,the disparity gradient limit principle is used in the present inventionto estimate the uncertainty of a predicted disparity for a particularpixel. That uncertainty is then used to define search ranges forcandidate pixel matches.

2. Least Commitment Strategy

A least commitment strategy states that only the most reliable decisionsshould be selected first and any unreliable decisions should bepostponed until enough confidence in the decision is accumulated.Because no irreversible decision is made (in other words, only reliabledecisions are made), this strategy offers significant flexibility inavoiding a locking search into a possibly incorrect step where anexpensive refinement (such as backtracking) would have to be exploited.

The present invention uses the least commitment strategy in thefollowing ways. First, the least commitment strategy is used by thepresent invention when defining a search range. If the match of a pixelhas to be searched in a wide range, there is a high probability that thefound match is not a correct one. It is preferable to defer matching ofthese pixels for as long as possible because the search range may bereduced later after more reliable matches are established.

Second, the least commitment strategy is used by the present inventionwhen search for matching pixels. In particular, a pixel is morediscriminating in a highly textured neighborhood and it is difficult todistinguish pixels in the same neighborhood having similar intensity.Thus, more reliable matches for pixels are found in areas with strongtextures, and the present invention attempts to match pixels in theseareas first.

Third, the present invention uses the least commitment strategy whendetermining whether a pixel has a match. Specifically, there may beseveral candidate matches for a pixel, and rather than using expensivetechniques (such as dynamic programming) to resolve the ambiguity, thepresent invention defers any decision. Once more reliable matches arefound in the future, the ambiguity will become lower because of a betterdisparity estimate having a smaller uncertainty.

Finally, the present invention uses the least commitment strategy toefficiently keep track of the status of pixel matches. If a pixel doesnot have any candidate match, there is a good possibility the pixel isoccluded by other pixels or is not in the camera's field of view. Inthis case, there is no need to search for a match for the pixel in thefuture. Similarly, if a match has already been found for the pixel,further search is not necessary. Both types of pixels are kept trackedby the present invention to increase efficiency and speed.

3. General Operation

In general, the present invention uses a progressive iterative processto determine a set of previous unambiguous pixel matches, define asearch range based on this set of previous unambiguous pixel matches,search for current unambiguous pixel matches within the search range andappend current unambiguous pixel matches to the set of previousunambiguous pixel matches. This iterative process of increasing the setof previous unambiguous pixel matches by appending current unambiguouspixel matches and redefining the search range based on the set ofprevious unambiguous pixel matches continues until suitable matching hasbeen achieved. The iterative process of the present inventionprogressively finds more and more unambiguous pixel matches anddynamically defines a search range based on these reliable pixel matchesto reduce potential false matches and increase reliability andefficiency.

FIG. 4 is a general flow diagram of the operation of the presentinvention. Initially, at least two images are received as input (box410). A set of previous unambiguous pixel matches then are determined(box 420). In general, if this is the first iteration the set ofprevious unambiguous pixel matches are obtained by finding a fewreliable pixel matches in order to begin an iteration matching process.Otherwise, if this is a subsequent iteration, a set of previousunambiguous pixel matches obtained from previous iterations generally isused. The set of previous unambiguous pixel matches is used to define asearch range (box 430). In particular, a size and an orientation of thesearch range is defined based on the set of previous unambiguous pixelmatches and determines the location where the present invention willsearch for additional pixel matches within each of the images.

As discussed further below, the correlation technique of the presentinvention includes a search range that may be rotated in an image topermit a more reliable matching process. Pixels within the search rangeare obtained from each image and a correlation between the pixels isperformed to compute a correlation score (box 440). Depending on thecorrelation score, a match label is assigned to the pixel (box 450).This match label specifies whether the pixel has been matched, notmatched, or is still unknown. By way of example, one set of match labels(that are used in the example below) are MATCHED, NOMATCH and UNKNOWN.

When pixels within the search range are matched these currentunambiguous pixel matches are appended to the set of previousunambiguous pixel matches (box 460). This updated set of previousunambiguous pixel matches is used in subsequent iterations to redefinethe search range (box 470). The present invention continues thisiterative process of progressively finding new unambiguous pixelmatches, adding them to the set of previous unambiguous pixel matchesand redefining the search range until the images have been suitablymatched. Suitably matched generally means, for example, that a maximumnumber of iterations have occurred or that no further pixel matches canbe found.

FIG. 5 is a general flow diagram of the operation of the search rangemodule shown in FIG. 3. In general, a pixel from a first image is aninput (box 510) and the disparity of the pixel is predicted (box 520).Next, an estimate is made concerning the uncertainty of the predicteddisparity (box 530). A determination is made as to whether the predicteddisparity is reliable (box 540) and, if not, a decision is deferred, andanother pixel is selected pixel and the matching iteration is performedagain. If the predicted disparity is reliable, the pixel is sent to thecorrelation module 330 to determine how well the pixel in the firstimage and the pixel in the second image correlate.

4. Detailed Operation

FIG. 6 is a detailed flow diagram illustrating a preferred embodiment ofthe present invention. It should be noted that the present invention maybe implemented in a variety of ways and actual implementation may varyfrom the following description. In this preferred embodiment, pixels inhighly textured areas are considered first. Textureness is measured asthe sample deviation of the intensity within a correlation window.Moreover, in order for a pixel in the first image to be considered itssample deviation must be larger than an intensity threshold value,T_(σ1) (called ThresholdSigmaIntensity). The ThresholdSigmaIntensity,T_(σ1), evolves with each iteration and is given by a monotonic functionthat never increases with iteration. Similarly, if a given pixel in thefirst image has a large uncertainty of its disparity vector, then in apreferred embodiment the pixel should be considered as late as possible.For a pixel to be considered, the standard deviation of its predicteddisparity vector must be smaller than a threshold T_(σD) (calledThresholdSigmaDisparity). The ThresholdSigmaDisparity, T_(σD), evolveswith each iteration and is given by a monotonic function that neverdecreases with iteration. Thus, only pixels having a good prediction ofthe disparity vector are considered at the beginning of an iteration.

As shown in FIG. 6, the present invention computes an intensitythreshold value and a disparity threshold value (box 600). Thesethreshold values may be determined in a suitable manner such as, forexample, by a user. In this preferred embodiment, the present inventiongives a pixel in a first image one the following match labels: (a)MATCHED (indicating the pixel has been matched); (b) NOMATCH (indicatingno pixel matches have been found); and (c) UNKNOWN (not yet decided).Initially, all pixels are given a match label of UNKNOWN, and a fewreliable pixel matches are obtained (box 605) using a reliabletechnique. Once a few seed pixel matches have been obtained, thematching process iteration begins.

An ambiguous pixel from a first image is input (box 610), labeled asUNKNOWN and a pixel disparity of that pixel is predicted as well as theuncertainty of that disparity prediction (box 615). Next, a list ofcandidate pixels in the second image is computed (box 620) thatsatisfies the epipolar constraint and disparity gradient limitconstraint. A normalized cross correlation, which is discussed in detailbelow, is used as a matching criterion. For a pair of pixels between twoimages, a normalized cross correlation score is computed (box 630)between two search ranges, one for each image. In a preferredembodiment, the search range uses two small windows (called correlationwindows) centered at the pixel being examined. The correlation scoreranges from −1, for two correlation windows that are not similar at all,to +1, for two correlation windows that are approximately identical.

A pair of pixels is considered as a potential match if their correlationscore is larger than the intensity threshold value. The list ofcandidate pixels is ordered on the epipolar line (box 650) and thecorrelation scores are plotted to form a correlation curve (box 660). Adetermination is then made as to whether two or more peaks of thecorrelation curve exceed the intensity threshold value (box 665). If so,then the pixel match is ambiguous (box 670) and, in accordance with theleast commitment strategy, the decision about the pixel match isdeferred to a later time and the pixel remains labeled as UNKNOWN.

If not, then a determination is made as to whether the correlation curveexceeds the intensity threshold value (box 675). If there is no peakexceeding the intensity threshold value then there is no pixel match(box 680) and the pixel is label as NOMATCH. Alternatively, if there isone peak on the correlation curve exceeds the intensity threshold valuethen the pixel at the peak is considered as the match of the given pixelin the first image (box 685) and the given pixel is labeled as MATCHED.Another iteration is performed in accordance with the present inventionand another pixel from the first image (labeled UNKNOWN) is selected formatching (box 690). The matching process of the present inventioncontinues this progressive iterative process until no more matches canbe found or the maximum number of iterations is attained.

The preferred embodiment of the present invention as shown in FIG. 6 hasa number of important properties. For example, the present inventionuses a progressive matching technique. Accordingly, the number of pixelsexamined in each iteration becomes progressively smaller. In addition,as discussed further below, a search range for a pixel is reduced whenthe disparity is updated with more matched pixels. The progressiveproperty of the present invention guarantees that the iterative matchingtechnique is actually making some progress and that the search space isbeing reduced.

The present invention also has the property of monotonicity. Forexample, in a preferred embodiment the present invention uses themonotonic functions ThresholdSigmaIntensity, T_(σ1) andThresholdSigmaDisparity, T_(σD). For each iteration, theThresholdSigmaIntensity, T_(σ1) is getting smaller and theThresholdSigmaDisparity, T_(σD) is getting larger. This means that theprobability that a pixel labeled as UNKNOWN is selected for a matchingtest becomes higher and eventually results in more pixels having MATCHEDor NOMATCH labels. Together with the up-date of disparity vectors andtheir uncertainty, the property of monotonicity guarantees that the setof UNKNOWN pixels considered is truly different from that prior torefinement. In other words, the set of UNKNOWN pixels is different inthe sense of the actual pixels considered and also of their candidatepixels to match in the other image.

The present invention also has the property of completeness, which meansthat adding more pixels labeled as MATCHED or NOMATCH will not lose anypotential matches. This property is desirable because it eliminates theneed to perform an expensive refinement procedure (such asbacktracking). Completeness is achieved because of the least commitmentstrategy, provided that the disparity gradient limit constraint issatisfied over the entire observed scene.

5. Correlation Technique

The present invention uses a novel correlation technique (including adynamic search range that permits rotation of a correlation window) tocorrelate pixels of one image to pixels of another image. In particular,a search range is determined that includes a correlation window capableof being rotated within an image. This rotation of the correlationwindow provides a more robust estimate of a correlation between pixels.

FIG. 7 illustrates the novel correlation technique between a first image700 and a second image 710 of the present invention. In particular,consider a pair of points m 715 and m′ 720 as shown in FIG. 7, where thecorresponding epipolar lines l 730 and l′ 740 are also drawn. AEuclidean transformation is computed where R(θ) is a 2D rotation matrixm′ _(i) =R(θ)(m _(i) −m)+m,  (10)with a rotation angle equal to θ, which is the angle between the twoepipolar lines 730, 740. This sends m 715 to m′ 720 and a point on l 730to a point on l′ 740.

A first correlation window 750 is chosen for the first image 700 that iscentered at m 715 and having one side parallel to the epipolar line l730. Similarly, a second correlation window 760 is chosen for the secondimage 710 that is centered at m′ 720. In order to have one side of thesecond correlation window 760 parallel to the epipolar line l′ 740 thecorrelation technique of the present invention allows rotation of thesecond correlation window 760. Although the shape of the first andsecond correlation windows 750, 760 are rectangular, it should be notedthat a window of any geometry could have been chosen.

A point m_(i) 770 within the first correlation window 750 corresponds toa point m′_(i) 780 within the second correlation window 760 as given byequation (10). Point m′_(i) 780 is generally not on the pixel grid, andits intensity is computed through bilinear interpolation from its fourneighboring pixels. A correlation score is then computed between pointm_(i) 770 in the first correlation window 750 and point m′_(i) 780 inthe second correlation window 760 according to equation (10). Anormalized cross correlation is used, which set the correlation scoreequal to 1 for two identical sets of pixels and equal to −1 for twocompletely different sets.

VI. Working Example and Exemplary Implementation

FIG. 8 is a working example of FIG. 4 whereby stereo matching of imagesrepresenting a scene is performed in accordance with the presentinvention. In this working example, a left image 800 and a right image810 of a scene are input images of the scene (box 820). Points withinthe left and right images 800, 810 are matched (box 830) using thestereo matching module 240 of FIG. 2. Matched points represent a 3-Dpoint in space and are used to determine 3-D structure information of ascene (box 840), which, for example, may be used to perform a 3-D scenereconstruction.

An exemplary implementation of the working example of FIG. 8 will now bediscussed. In this exemplary implementation, the present invention isformulated within a Bayesian inference framework. Bayesian inference,which is well-known to those skilled in the art, is discussed in detailby D. S. Sivia in “Data Analysis: a Bayesian tutorial” (OxfordUniversity Press, 1996), the entire contents of which are herebyincorporated by reference. In this working example using a Bayesianinference framework, a first term, p(d|m,B), corresponds to a priordensity distribution of the disparity, where d is a disparity of thegiven pixel m, and B denotes relevant background information (such as anepipolar geometry and a set of matched pixels). A second termcorresponds to a sampling distribution p(I′|d,m,B), which includes thelikelihood of observed data (within a second image I′) when given d, mand B. Bayes rule is then used to combine the information in the datawith the prior probability, which yields a posterior densitydistribution, $\begin{matrix}{{{p\left( {\left. d \middle| I^{\prime} \right.,m,B} \right)} = \frac{{p\left( {\left. I^{\prime} \middle| d \right.,m,B} \right)}{p\left( {\left. d \middle| m \right.,B} \right)}}{p\left( {\left. I^{\prime} \middle| m \right.,B} \right)}},} & (1)\end{matrix}$where p(I′|m,B) does not depend on d and can be considered constantbecause the second image I′ is fixed. Accordingly, the factor p(I′|m,B)can be omitted and an unnormalized posterior density distribution,p(I′|d,m,B) p(d|m,B), is considered (denoted by p(d|I′,m,B)).Computations that summarize p(d|I′,m,B) are performed and a pixel isconsidered to determine whether the pixel should be classified asMATCHED or NOMATCH, or classified as UNKNOWN for a future decision.1. Prediction of Pixel Disparity and Uncertainty

As discussed above, the present invention uses a novel correlationtechnique (including a dynamic search range capable of rotation) toconstrain a search for potential pixel matches based on a disparitygradient limit. The disparity gradient limit is exploited to estimate anuncertainty of a predicted disparity for a particular pixel, and thisuncertainty is used to dynamically define a search range for candidatepixel matches. In this working example, the disparity gradient limitprinciple is used to predict pixel disparity and uncertainty as follows.

Disparity is well defined for parallel cameras where the two imageplanes are identical. In this case, without any loss of generality, anassumption may be made that the horizontal axis in both images isaligned. Given a first pixel of coordinates (u,v) in a first image and acorresponding second pixel of coordinates (u′,v′) in a second image,disparity is defined as the difference d=v′−v. Disparity is inverselyproportional to the distance of the 3-D point to the cameras. Thus, adisparity of zero implies that the 3-D point is at infinity.

Consider two 3-D points whose projections are m₁=[u₁,v₁]^(T) and m₂=[u₂,v₂]^(T) in the first image, and m′₁=[u′₁,v′₁]^(T) and m′₂=[u′₂,v′₂]^(T)in the second image (where u′₁=u₁ and u′₂=u₂ when the two camera areparallel). The disparity gradient of these two points is defined to bethe ratio of their difference in disparity to their distance in thecyclopean image. For example, for a pair of pixels in correspondencewith coordinates (u, v) and (u′, v′) the cyclopean image point is at((u+u′)/2, (v+v′)/2). In the first image, the disparity gradient isgiven by $\begin{matrix}{{D\quad G} = {{\frac{d_{2} - d_{1}}{v_{2} - v_{1} + {\left( {d_{2} - d_{1}} \right)/2}}}.}} & (2)\end{matrix}$

Experiments in psychophysics have shown that human perception imposes aconstraint that the disparity gradient DG is upper-bounded by a limit K.In other words, if a point on an object is perceived, neighboring pointshaving DG>K are not perceived correctly. For example, a theoreticallimit for opaque surfaces is K=2 to ensure that the opaque surfaces arevisible to both eyes. Although the range of allowable surfaces is largewith K=2, disambiguating power is weak because false matches receive andexchange as much support as correct ones. In another example (and anextreme limit), K≈0, which allows only nearly front-parallel surfaces.In the PMF algorithm described by S. B. Pollard, J. E. W. Mayhew and J.P. Frisby entitled “PMF: a stereo correspondence algorithm using adisparity gradient constraint” in Perception (14: 449-470, 1985), theentire contents of which are hereby incorporated by reference, thedisparity gradient limit K is a free parameter, which can be varied overrange from 0 to 2. An intermediate value (such as a value between 0.5and 1) allows selection of a convenient tradeoff point between allowablescene surface jaggedness and disambiguating power because most falsematches produce relatively high disparity gradients.

When the cameras are in “general position” (that is, image planes of thecameras do not lie on the same plane), it is not reasonable to definescalar disparity as a simple function of the image coordinates of twopixels in correspondence. In this working example, a disparity vectord=[u′−u, v′−v]^(T) is used. In general terms, this disparity vector isthe same as a flow vector used in optical flow computation. If a scalarvalue is necessary, we use d=∥d∥ and call the scalar value thedisparity. If objects are observed that are smooth almost everywhere,both the disparity vector (d) and the disparity (d) should varysmoothly. Similar to equation (2), for two points m₁ and m₂ in the firstimage, we define the disparity gradient as, $\begin{matrix}{{D\quad G} = {\frac{{d_{2} - d_{1}}}{{m_{2} - m_{1} + {\left( {d_{2} - d_{1}} \right)/2}}}.}} & (3)\end{matrix}$

Imposing a gradient limit constraint that DG≦K yields,∥d ₂ −d ₁ ∥≦K∥m ₂ −m ₁+(d ₂ −d ₁)/2∥.Using an inequality ∥v₁+v₂∥≦∥v₁∥+∥v₂∥ for any vectors v₁ and v₂, gives,∥d ₂ −d ₁ ∥≦K∥m ²⁻ m ₁ ∥+K∥(d ₂ −d ₁)/2∥which leads immediately, for K<2, to, $\begin{matrix}{{{{d_{2} - d_{1}}} \leq {\frac{2K}{2 - K}D}},} & (4)\end{matrix}$where D=∥m₂−m₁∥ is the distance between m₁ and m₂. This leads to thefollowing result:

-   -   Lemma 1: Given a pair of matched points (m₁, m′₁) and a point m₂        in the neighborhood of m₁, the corresponding point m′₂ that        satisfies the disparity gradient constraint with limit K must be        inside a disk centered at m₂+d₁ with radius equal to        ${\frac{2K}{2 - K}D},$        which is called the continuity disk.        In other words, in absence of other knowledge, the best        prediction of the disparity of m₂ is equal to d₁ with the        continuity disk defining the uncertainty of the disparity        prediction.

In some cases it may be desirable to have the actual disparity at thecentral part of the continuity disk. Further, it may desirable toconsider a small probability that the actual disparity is outside of thecontinuity disk (due to, for example, an occlusion or a surfacediscontinuity). Therefore, in this working example, the uncertainty ismodeled as an isotropic Gaussian distribution with a standard deviationequal to half of the radius of the continuity disk. More precisely,given a pair of matched points (m_(i), m′_(i)), the disparity of a pointm is modeled asd=d _(i) +D _(i) n _(i),  (5)where d_(i)=m′_(i)−m_(i), D_(i)=|m−m_(i)∥, and n_(i)˜N(0, σ_(i) ²I) withσ_(i)=K/(2−K). Note that disparity d_(i) also has its own uncertaintydue to limited image resolution. The density distribution of d_(i) isalso modeled in this working example as a Gaussian, namely,p(d_(i))=N(d_(i)|{overscore (d)}_(i), σ_(d) _(i) ²I). Thus, the densitydistribution of disparity d is given byp(d|(m _(i) ,m′ _(i)),m)=N(d|{overscore (d)} _(i),(σ_(d) _(i) ² +D _(i)²σ_(i) ²)I).  (6)

Given a set of point matches {(m_(i),m′_(i))|i=1, . . . ,n}, equation(6) then provides n independent predictions of disparity d. The priordensity distribution of the disparity, p(d|m, B), can be obtained bycombining these predictions with a minimum variance estimator,p(d|m,B)=N(d|{overscore (d)},ν ² I),  (7)where $\begin{matrix}{\overset{\_}{d} = {\left( {\sum\limits_{i = 1}^{n}\frac{1}{\sigma_{d_{i}}^{2} + {D_{i}^{2}\sigma_{i}^{2}}}} \right)^{- 1}{\sum\limits_{i = 1}^{n}{\frac{1}{\sigma_{d_{i}}^{2} + {D_{i}^{2}\sigma_{i}^{2}}}{\overset{\_}{d}}_{i}}}}} \\{\sigma^{2} = {\left( {\sum\limits_{i = 1}^{n}\frac{1}{\sigma_{d_{i}}^{2} + {D_{i}^{2}\sigma_{i}^{2}}}} \right)^{- 1}.}}\end{matrix}$Further, this may be made more robust by first identifying the Gaussianwith smallest variance and then combining it with those Gaussians whosemeans fall within two or three standard deviations.

Next, σ_(i) (which as mentioned above is related to the disparitygradient limit K) is chosen. Considering that the disparity gradientconstraint is still a local one, it should become less restrictive whena point being considered is away from a matched point. Hence, a range[σ_(min),σ_(max)] is specified and σ_(i) is given byσ_(i)=(σ_(max)−σ_(min))(1−exp(−D _(i) ²/τ²))+σ_(min.).  (8)When D_(i)=0, σ_(i)=σ_(min), and when D_(i)=χ, σ_(i)=σ_(max). Theparameter τ in equation (8) controls how fast the transition fromσ_(min) to σ_(max) is expected. In this working example, σ_(min)=0.3pixels, σ_(max)=1.0 pixel, and τ=30. This is equivalent to K_(min)=0.52and K_(max)=1.34.2. Computation of Pixel Disparity Likelihood

The sampling distribution, p(I′|d,m,B), is computed next to obtain alikelihood that, given d, m and B, a pixel in the second image I′ is amatch for a first pixel. Because of the epipolar constraint, there is noneed to compute the density for each pixel in I′. Further, there is alsono need to compute the density for each pixel on the epipolar line of mbecause the prior density has been computed in equation (7). A list ofpixels of interest (called candidate pixels and denoted by Q(m)) is anintersection of the epipolar line of m with a continuity disk as definedin Lemma 1.

Densities are related to correlation scores C_(j) between m in the firstimage and each candidate pixel m′_(j)εQ(m) in the second image. Usingthe correlation technique of the present invention as described above, acorrelation score C that is approximately equal to −1 means that pixelsare not similar at all while a correlation score C that is approximatelyequal to +1 means that pixels are generally identical. Correlationscores are mapped to densities by adding 1 followed by a normalization.More precisely, the correlation score C_(j) of a pixel m′_(j) isconverted into a density as $\begin{matrix}{{{p\left( {\left. {I^{\prime}\left( m_{j}^{\prime} \right)} \middle| d^{(j)} \right.,m,B} \right)} = \frac{C_{j} + 1}{\sum\limits_{k \in {Q{(m)}}}\left( {C_{j} + 1} \right)}},} & (9)\end{matrix}$where d^((j))=m′_(j)−m.3. Match Label Inference

Using a posterior density distribution, an inference can be made as tothe type of match between two pixels. In this working example, a matchlabel is assigned to a pixel based on this inference. Namely, asdiscussed above, a pixel match may be assigned a label of MATCHED,NOMATCH or UNKNOWN. In this working example, the posterior densitydistribution p(d|I′,m, B) is a multiplication of p(I′(m′_(j))|d^((j)),m, B) in equation (9) with p(d^((j))|m,B) in equation (7) for eachcandidate pixel m′_(j).

Based on the posterior density distribution, p(d|I′, m, B), a number ofinferences may be made. For example, if there is only one prominent peak(i.e., the peak is above a certain threshold value), the probabilitythat this is a correct match is very high, and thus the pixel in thefirst image is given a match label of MATCHED. If there are two or moreprominent peaks above the threshold, the matching ambiguity is high (inother words, the probability of making a wrong decision is high). Usingthe least commitment principle, this pixel is given a match label ofUNKNOWN and is left to evolve. If there is no prominent peak at allabove the threshold, the probability that the corresponding pixel in thesecond image is not visible is very high (meaning that the pixel may beeither occluded by other pixels or out of the field of view). In thiscase, the pixel in the first image is given a match label of NOMATCH.

In order to facilitate the task of choosing an appropriate threshold onthe posterior density distribution, and because the posterior densitydistribution otherwise unnormalized, the prior and likelihood functionsare normalized differently. The prior function in equation (7) ismultiplied by σ{square root}{square root over (2π)} so that its maximumvalue is equal to one. The likelihood function in equation (9) ischanged to (C_(j)+1)/2 so that it is equal to 1 for identical pixels and0 for completely different pixels. Thus, in this working example, a peakin the posterior density distribution is considered a prominent peak(i.e., above a threshold value) if its value is greater than 0.3. Forexample, this corresponds to a situation where C_(j)=0.866 and thedisparity is at 1.5σ.

4. Results

In the working example of FIG. 8 (using the above exemplaryimplementation) a scene that included an office scene with books and atelephone on a desk was used. Although the images in the office scenewere recorded in color, only black and white information was used. Theimage resolution for the office scene was 740×480 pixels.

In order to reduce computation cost, instead of using all previouslyfound matches in predicting disparities and their uncertainties, onlythree neighboring points defined by a Delaunay triangulation were used.In particular, a dynamic Delaunay triangulation technique was usedbecause of its efficiency in updating the triangulation when more pointmatches are available. This technique is discussed in detail by O.Devillers, S. Meiser and M. Teilluad in “Fully dynamic Delaunaytriangulation in logarithmic expected time per operation” (Comput. Geom.Theory Appl., 2 (2): 55-80, 1992), the entire contents of which arehereby incorporated by reference. It is reasonable to use only threeneighboring points because other points are usually much farther away,resulting in a larger uncertainty in its predicted disparity, hencecontributing little to the combined prediction of the disparity given inequation (7).

Initial point matches (together with the fundamental matrix) wereobtained automatically using a robust image matching technique. Thismatching technique is described by Z. Zhang, R. Deriche, O. Faugeras andQ. T. Luong in “A robust technique for matching two uncalibrated imagesthrough the recovery of the unknown epipolar geometry” (ArtificialIntelligence Journal, 78: 87-119, October 1995), the entire contents ofwhich are hereby incorporated by reference. A search range of [−60,60](pixels) for both horizontal and vertical directions was used for thescene.

Table 1 illustrates the number of pixel matches after each iteration.The number of matches for iteration 0 indicates the number of initialmatches found by the robust matching algorithm. Note that instead ofworking on each pixel, only one every four pixels actually wasconsidered because of the memory limitation in the Delaunaytriangulation algorithm. Further, the values of the functionsThresholdSigmaIntensity, T_(σ), and ThresholdSigmaDisparity, T_(σ) _(D)with respect to the iteration number are given in the second and thirdrows of Table 1. For example, for iteration 4, T_(σ) ₁ =4 and T_(σ) _(D)=18. TABLE 1 Number of matched pixels in each iteration Iteration 0 1 23 4 5 6 T_(σ) _(I) 7 6 5 4 3 2 T_(σ) _(D) 12 14 16 18 20 20 Number 141455 712 939 1239 1440 1500 of Pixel Matches

After the initial set of point matches was obtained, the disparity andits uncertainty were predicted for each iteration after iteration 0. Inparticular, disparity vectors were displayed for every 10 pixels andtheir lengths were half of their actual magnitudes. In addition, thestandard deviation of the predicted disparities was calculated. FIGS.9A-9D illustrate the standard deviation of the predicted disparities ofthe working example for various iterations. Each figure is shown in graylevels after having been multiplied by 5 and truncated at 255. Thus, inFIGS. 9A-9D, a “black” pixel in a figure means that the predicteddisparities were quite reliable, while a “white” pixel implies that thepredicted disparities were very uncertain.

FIG. 9A illustrates the predicted disparity for the images shown in FIG.8 after the initial point matches. A dark area 900 is the area where thepredicted disparities were quite reliable and a light area 910 is thearea where the predicted disparities were uncertain. It should be notedin FIG. 9A that the dark area 900 and the light area 910 areapproximately equal in size. FIG. 9B illustrates the predicted disparityafter the second iteration, where a dark area 920 has grown to be largerthan a light area 930, meaning that there is a larger number of pixelshaving reliable predicted disparities. FIG. 9C, which illustrates thepredicted disparity after the fourth iteration, shows dark area 940 muchlarger than a light area 950. Further, FIG. 9D illustrates the predicteddisparity after the sixth iteration and shows a dark area 960 evenlarger still than a light area 970, with the dark area 960 nearlyengulfing the entire image. Clearly, it can be seen how quickly reliablepixel matches evolve using the present invention. In particular, asshown in FIGS. 9A-9D, the uncertainty image becomes darker quickly,indicating that the predicted disparity of the pixel matches is becomingincreasingly reliable after only a few iterations.

Once the stereo matching of the images in FIG. 8 was performed inaccordance with the present invention, a 3-D Euclidean reconstructionwas obtained. FIG. 10 illustrates a view of the 3-D Euclideanreconstruction 1000 using 3-D structure information obtained inaccordance with the present invention. It should be noted that theoffice scene (especially the book structure 1010) has been preciselyrecovered.

The foregoing description of the preferred embodiments of the inventionhas been presented for the purposes of illustration and description. Itis not intended to be exhaustive or to limit the invention to theprecise form disclosed. Many modifications and variations are possiblein light of the above teaching. It is intended that the scope of theinvention be limited not by this detailed description of the invention,but rather by the claims appended hereto.

1. A method of stereo matching digital images representing a scene,comprising: determining possible pixel matches between a target pixelfrom a first digital image and pixels from a second digital image;evaluating a reliability of each of the possible pixel matches;predicting a disparity between the target pixel and pixels from thesecond image; estimating an uncertainty of the predicted pixeldisparity; and classifying a certain pixel match based on thereliability of the possible pixel matches.
 2. The method of claim 1,wherein the pixels from the second image satisfy an epipolar constraint.3. The method of claim 2, wherein the pixels from the second imagesatisfy a disparity gradient limit constraint.
 4. The method of claim 1,wherein the reliability is evaluated using the least commitmentstrategy.
 5. (canceled)
 6. The method of claim 1, wherein classifying acertain pixel match comprises: generating a list of candidate pixelsfrom the pixels of the second image; ordering the list of candidatepixels on an epipolar line; determining a correlation score for eachpixel on the list of candidate pixels; and computing a correlation curvefrom the correlation scores. 7-10. (canceled)
 11. A computer-readablemedium having computer-executable instructions for performing the methodrecited in claim
 1. 12. A method of stereo matching digital imagesrepresenting a scene, comprising: determining possible pixel matchesbetween a target pixel from a first digital image and pixels from asecond digital image; evaluating a reliability of each of the possiblepixel matches using a least commitment strategy; and classifying acertain pixel match based on the reliability of the possible pixelmatches.
 13. The method of claim 12, wherein the pixels from the secondimage satisfy an epipolar constraint.
 14. The method of claim 13,wherein the pixels from the second image satisfy a disparity gradientlimit constraint.
 15. The method of claim 12, wherein evaluating thereliability comprises: predicting a disparity between the target pixeland pixels from the second image; and estimating an uncertainty of thepredicted pixel disparity.
 16. The method of claim 12, whereinclassifying a certain pixel match comprises: generating a list ofcandidate pixels from the pixels of the second image; ordering the listof candidate pixels on an epipolar line; determining a correlation scorefor each pixel on the list of candidate pixels; and computing acorrelation curve from the correlation scores.
 17. A computer-readablemedium having computer-executable instructions for performing the methodrecited in claim
 12. 18. A method of stereo matching digital imagesrepresenting a scene, comprising: determining possible pixel matchesbetween a target pixel from a first digital image and pixels from asecond digital image; evaluating a reliability of each of the possiblepixel matches; classifying a certain pixel match based on thereliability of the possible pixel matches; generating a list ofcandidate pixels from the pixels of the second image; ordering the listof candidate pixels on an epipolar line; determining a correlation scorefor each pixel on the list of candidate pixels; and computing acorrelation curve from the correlation scores.
 19. The method of claim18, wherein the pixels from the second image satisfy an epipolarconstraint.
 20. The method of claim 19, wherein the pixels from thesecond image satisfy a disparity gradient limit constraint.
 21. Themethod of claim 18, wherein the reliability is evaluated using the leastcommitment strategy.
 22. The method of claim 18, wherein evaluating thereliability comprises: predicting a disparity between the target pixeland pixels from the second image; and estimating an uncertainty of thepredicted pixel disparity.
 23. A computer-readable medium havingcomputer-executable instructions for performing the method recited inclaim 18.